Question: Assume that a production function has the following form,Upper Y equals AK Superscript alpha Baseline Upper N Superscript betaY=AKN, wherealpha plus beta equals 1+=1and A=1.If
Assume that a production function has the following form,Upper Y equals AK Superscript alpha Baseline Upper N Superscript betaY=AKN, wherealpha plus beta equals 1+=1and A=1.If we wanted to algebraically demonstrate that this production function has constant returns to scale, which of the following would represent the first two steps of this process?A.Upper F left parenthesis 2 Upper K comma 2 Upper N right parenthesis equals left parenthesis 2 Upper K right parenthesis Superscript 1 Baseline left parenthesis 2 Upper N right parenthesis Superscript 1F(2K,2N)=(2K)1(2N)1and Upper F left parenthesis 2 Upper K comma 2 Upper N right parenthesis equals 2 Upper K times 2 Upper NF(2K,2N)=2K2NB.Upper F left parenthesis Upper K comma 2 Upper N right parenthesis equals left parenthesis Upper K right parenthesis Superscript alpha Baseline left parenthesis 2 Upper N right parenthesis Superscript betaF(K,2N)=(K)(2N)andUpper F left parenthesis Upper K comma 2 Upper N right parenthesis equals Upper K Superscript alpha Baseline times 2 Superscript beta Baseline times Upper N Superscript betaF(K,2N)=K2NC.Upper F left parenthesis 2 Upper K comma Upper N right parenthesis equals left parenthesis 2 Upper K right parenthesis Superscript alpha Baseline left parenthesis Upper N right parenthesis Superscript betaF(2K,N)=(2K)(N)and Upper F left parenthesis 2 Upper K comma Upper N right parenthesis equals 2 Superscript alpha Baseline times Upper K Superscript alpha Baseline times Upper N Superscript betaF(2K,N)=2KND.Upper F left parenthesis 2 Upper K comma 2 Upper N right parenthesis equals left parenthesis 2 Upper K right parenthesis Superscript alpha Baseline left parenthesis 2 Upper N right parenthesis Superscript 1 minus alphaF(2K,2N)=(2K)(2N)1andUpper F left parenthesis 2 Upper K comma 2 Upper N right parenthesis equals 2 Superscript alpha Baseline times Upper K Superscript alpha Baseline times 2 Superscript 1 minus alpha Baseline times Upper N Superscript 1 minus alphaF(2K,2N)=2K21N1When you have completed this process the right hand side of the equation will equalUpper K Superscript alpha Baseline Upper N Superscript betaKN.Part 2If we write the original production function in the form ofStartFraction Upper Y Over Upper N EndFraction equals Upper F left parenthesis StartFraction Upper K Over Upper N EndFraction comma 1 right parenthesisYN=FKN,1,it will become:A.StartFraction Upper Y Over Upper N EndFraction equals Upper A left parenthesis StartFraction Upper K Over Upper N EndFraction right parenthesis Superscript alphaStartFraction Upper Y Over Upper N EndFraction equals Upper A left parenthesis StartFraction Upper K Over Upper N EndFraction right parenthesis Superscript alphaYN=AKNB.StartFraction Upper Y Over Upper N EndFraction equals Upper A StartFraction Upper K Superscript alpha Over left parenthesis Upper N plus 1 right parenthesis Superscript beta EndFractionStartFraction Upper Y Over Upper N EndFraction equals Upper A StartFraction Upper K Superscript alpha Over left parenthesis Upper N plus 1 right parenthesis Superscript beta EndFractionYN=AK(N+1)C.StartFraction Upper Y Over Upper N EndFraction equals Upper A left parenthesis StartFraction Upper K Over Upper N EndFraction plus 1 right parenthesisStartFraction Upper Y Over Upper N EndFraction equals Upper A left parenthesis StartFraction Upper K Over Upper N EndFraction plus 1 right parenthesisYN=AKN+1D.StartFraction Upper Y Over Upper N EndFraction equals AK Superscript alpha plus 1StartFraction Upper Y Over Upper N EndFraction equals AK Superscript alpha plus 1
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